Theory of the N-Body Problem

June 9, 1996

29

Marciniak's book

(12:72)

, his 7th order Adam-Bashford method gave much worse results

than his 4th order Adam-Bashford method in a particular case. In my testing I found a case

where the 4th order Adam-Bashford method with an accuracy parameter of one (i.e. the

command `xstar -m ab4 -a 1') gave much better results than the 7th order Adam-Bashford

method with the higher accuracy parameter of -a 4. It also gave better results than -m ab4

with the higher accuracy parameter -a 2. Obviously, sometimes a method just gets lucky

(or unlucky).

One more point should be made when estimating which method of integrating an

ODE might be better. Each time the derivative of

*f()*

is taken, a constant is pulled out due

to the inverse square property of

*f().*

So the first derivative of

*f()*

cause the constant -2 to be

pulled out, the second derivative pulls out the constant -3. The constant of the error term

of a 7th order method has to be multiplied by -9! = -362880. This is a very large constant

that the higher order method has to over come and one of the reasons why lower order

methods are more accurate when lower accuracy parameter settings are used.

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